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Properties

Label 1-2475-2475.1091-r0-0-0
Degree $1$
Conductor $2475$
Sign $0.316 + 0.948i$
Analytic cond. $11.4938$
Root an. cond. $11.4938$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.669 + 0.743i)7-s + (−0.809 + 0.587i)8-s + (0.104 + 0.994i)13-s + (0.5 − 0.866i)14-s + (0.669 − 0.743i)16-s + 17-s + (0.809 − 0.587i)19-s + (0.978 + 0.207i)23-s + (−0.309 − 0.951i)26-s + (−0.309 + 0.951i)28-s + (−0.104 + 0.994i)29-s + (−0.978 − 0.207i)31-s + (−0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.669 + 0.743i)7-s + (−0.809 + 0.587i)8-s + (0.104 + 0.994i)13-s + (0.5 − 0.866i)14-s + (0.669 − 0.743i)16-s + 17-s + (0.809 − 0.587i)19-s + (0.978 + 0.207i)23-s + (−0.309 − 0.951i)26-s + (−0.309 + 0.951i)28-s + (−0.104 + 0.994i)29-s + (−0.978 − 0.207i)31-s + (−0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.316 + 0.948i$
Analytic conductor: \(11.4938\)
Root analytic conductor: \(11.4938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2475,\ (0:\ ),\ 0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8136567095 + 0.5861837039i\)
\(L(\frac12)\) \(\approx\) \(0.8136567095 + 0.5861837039i\)
\(L(1)\) \(\approx\) \(0.7035491088 + 0.1855082701i\)
\(L(1)\) \(\approx\) \(0.7035491088 + 0.1855082701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.978 + 0.207i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (0.978 + 0.207i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.669 + 0.743i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (0.913 - 0.406i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.34174084444413359092514239375, −18.62284923982387273003762806958, −17.977394907542203594099518223968, −17.19287542747005049077819001295, −16.5551183381536950792734642515, −16.0948398078583739872289007219, −15.19453930885822394599058316034, −14.477632058164820415502098779412, −13.39624250849754473935110255534, −12.75852140069816191526392138125, −12.07675308719706086565294540612, −11.13076522454929018636039750305, −10.52574798546170054919928364403, −9.80989213997620289483468091151, −9.35640673147095703027981150688, −8.241867076563926456900755791968, −7.64216378365408614545352899693, −7.05662644763372517732717863243, −6.08406198846429700989411294187, −5.39557453820892653085362438843, −3.969146311291067394788646776245, −3.29200535153157573096702945509, −2.587437086662863165633160402107, −1.27807873821075684341159209902, −0.616113820398936630596852705980, 0.89973364047648892322485242157, 1.84543637402528312389344520467, 2.81993145817118850392050649271, 3.46981638048160905129053747093, 4.913250675042523075180766061155, 5.67639218151868654877039059678, 6.443234009432021064806600107271, 7.15279468503861759332638612781, 7.87143226195293212892028676922, 8.85317223072483141911450395107, 9.44136751851364290955415572677, 9.76540511066271144160144671662, 11.101441138759056700362495824478, 11.32719872890256100643208473418, 12.39357797621158355534658652679, 12.912882353255127528379150078637, 14.223721445012455331468414407646, 14.651844354125649903555723314559, 15.6496898178472386154183369035, 16.147817349457274372623530630702, 16.69585655462743828853117109362, 17.51837838123221378833293769996, 18.3872209678755568924118455858, 18.7690290730361993568574088041, 19.4468391589053848278111917564

Graph of the $Z$-function along the critical line